Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (3pq)q²/6p²q⁴

Verified step by step guidance

1. Start by writing the expression clearly: \(\frac{(3pq)q^2}{6p^2q^4}\).
2. Distribute the exponent on the numerator: \((3pq)q^2 = 3pq \cdot q^2 = 3p q^{1+2} = 3p q^3\).
3. Rewrite the expression as \(\frac{3p q^3}{6 p^2 q^4}\).
4. Simplify the coefficients: \(\frac{3}{6} = \frac{1}{2}\).
5. Simplify the variables by subtracting exponents in the denominator from those in the numerator: \(p^{1 - 2} = p^{-1}\) and \(q^{3 - 4} = q^{-1}\). Then rewrite without negative exponents as \(\frac{1}{2} \cdot \frac{1}{p} \cdot \frac{1}{q}\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to multiply, divide, and simplify expressions with powers. Key rules include adding exponents when multiplying like bases and subtracting exponents when dividing. Understanding these rules is essential to simplify expressions like (3pq)q^2 / (6p^2q^4).

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the numerator and denominator by canceling common factors. This process requires factoring expressions and applying exponent rules to combine like terms, ensuring the expression is in its simplest form without negative exponents.

Negative Exponents and Their Elimination

Negative exponents indicate reciprocal powers, such as a^(-n) = 1/a^n. To write answers without negative exponents, rewrite terms with negative powers as fractions with positive exponents. This step is crucial for final answers in simplified expressions.